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Former Saracens chief executive Griffiths had only held the position since January 1 this year.
He was initially drafted in by Bath on a consultancy basis last November to lead an in-season review of the club’s rugby programme.
Bath are currently 12th in the Premiership with one game left, having lost 17 of their 23 league fixtures this term.
Those defeats included a 71-17 reversal at home to Saracens and a 64-0 loss at west country rivals Gloucester.
Bath owner Bruce Craig, who relinquished his role as chairman in January, said: “We thank Edward for all he has done in helping our club to refocus.
“He has brought expertise, knowledge and experience, and he leaves the club in a significantly stronger position.
“I will continue to support the club as we move forward.”
Bath said in a statement that Griffiths had “completed his term as chairman.”
The club added that he had been “contracted to lead Bath Rugby’s performance reset, and many of his recommendations have now been implemented.”
Current Munster boss Johann Van Graan will begin his role as Bath’s head of rugby on July 1.
Rugby director Stuart Hooper, meanwhile, takes on a job of general manager for next season.
Bath added: “Johann Van Graan will take on the role of head of rugby on 1st July, leading the club’s rugby programme.
“He assumes complete responsibility for rugby matters. Van Graan will concentrate on the coaching and performances of the team and will oversee all coaching and performance staff.
“Stuart Hooper assumes the new role of general manager, reporting to Van Graan.
“In his new role, Hooper will support Van Graan in delivering the club’s long-term plans, which include the further enhancement of its player pathway and academy, and the development of competitive advantage in areas of data, psychology, nutrition, training and performance facilities.”